Density fluctuations on mm and Mpc scales
N.P. Basse
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 21 January 2005; accepted 14 April 2005
Available online 21 April 2005 Communicated by F. Porcelli
Abstract
We will in this Letter report on suggestive similarities between density fluctuation power versus wavenumber on small (mm) and large (Mpc) scales. The small scale measurements were made in fusion plasmas and compared to predictions from classical fluid turbulence theory. The data is consistent with the dissipative range of 2D turbulence. Alternatively, the results can be fitted to a functional form that cannot be explained by turbulence theory. The large scale measurements were part of the Sloan Digital Sky Survey galaxy redshift examination. We found that the equations describing fusion plasmas also hold for the galaxy data. The comparable dependency of density fluctuation power on wavenumber in fusion plasmas and galaxies might indicate a common origin of these fluctuations.
2005 Elsevier B.V. All rights reserved.
PACS: 52.25.Fi; 52.35.Ra; 98.80.Bp; 98.80.Es
Keywords: Density fluctuations; Wavenumber spectra; Fusion plasmas; Galaxies; Turbulence
1. Introduction
If one were to make a survey of where we are, what we know and what we do not know about magneti- cally confined fusion plasmas, turbulence would cer- tainly be an area marked ‘Here Be Monsters’. Cross- field transport (perpendicular to the main magnetic field) assuming that only binary particle collisions contribute is called neoclassical transport [1]. This
E-mail address:basse@psfc.mit.edu(N.P. Basse).
URL:http://www.psfc.mit.edu/people/basse/.
transport level includes effects associated with toroidal geometry. However, in general the measured transport is several orders of magnitude larger than the neoclas- sical one, especially for electrons. This phenomenon has been dubbed anomalous transport and is subject to intense studies on most experimental fusion de- vices[2]. Anomalous transport is believed to be driven by turbulence in the plasma.
It is generally thought that turbulence creates fluc- tuations visible in most plasma parameters. Therefore a concerted effort has been devoted to the study of fluctuations and their relation to the global (and local) plasma confinement quality.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2005.04.033
In this Letter we study electron density fluctua- tion power versus wavenumber (also known as the wavenumber spectrum) in the Wendelstein 7-AS (W7- AS) stellarator[3]. Wavenumber spectra characterize the nonlinear interaction between turbulent modes.
The fluctuations were measured using the localized turbulence scattering (LOTUS) diagnostic[4,5].
As we shall see, the density fluctuation power P decreases exponentially with increasing wavenumber kon mm scales in fusion plasmas
P (k)∝ 1 (1)
k ×e−nk,
wheren >0 is a constant having a dimension of length and k=2π/λ, where λ is the corresponding wave- length. This was initially noted using the simplified form
(2) P (k)∝e−nk
in Ref.[6]. Eq.(2)also holds for density fluctuations in the Tore Supra tokamak[7].
Having the exponential structure of the wavenum- ber spectrum in mind, we were intrigued to see a figure in KVANT[8], a magazine published by the Danish Physical Society, showing the root-mean-square den- sity fluctuation amplitude of galaxies from the Sloan Digital Sky Survey (SDSS) [9] versus distance that seemed to display the same exponential behavior as we found in fusion plasmas. This remarkable similar- ity prompted us to apply the same analysis to the SDSS data as we had previously used for our fusion plasma measurements.
The Letter is organized as follows: In Section2we summarize our wavenumber spectrum measurements in fusion plasmas. Thereafter we describe inflation and the SDSS wavenumber spectrum in Section3. We dis- cuss the results in Section4and finally state our con- clusions in Section5.
2. Fusion plasmas
A wavenumber spectrum of turbulence in W7-AS is shown inFig. 1. The measured points are shown along with two power-law fits
(3) P (k)∝k−m,
wheremis a dimensionless constant. The power-law fits are shown as solid lines and an exponential fit to Eq. (1)is shown as a dashed line. The power-law fits are motivated by classical fluid turbulence the- ory where one expects wavenumber spectra to exhibit power-law behavior with exponents mdepending on the dimension of the observed turbulence:
• 3D: Energy is injected at a large scale and re- distributed (cascaded) by nonlinear interactions down to a small dissipative scale. In this case, the energy spectrum in the inertial rangeE(k)∝ k−5/3[10].
• 2D: Here, two power-laws exist on either side of the energy injection scale. For smaller wavenum- bers, the inverse energy cascade obeys E(k)∝ k−5/3and for larger wavenumbers, the enstrophy cascade followsE(k)∝k−3[11].
• 1D: Energy is injected at a large scale and dissi- pated at a small scale;E(k)∝k−2[12].
Our measured power is equivalent to the d-dimen- sional energy spectrumFd(k)[10,13,14]
P (k)=Fd(k)=E(k) Ad ,
(4) A1=2, A2=2π k, A3=4π k2,
where Ad is the surface area of a sphere having ra- dius k and dimension d. Usually on would assume that d =2 in fusion plasmas, since transport along magnetic field lines is nearly instantaneous. The fits to Eq. (3) in Fig. 1 yield exponents m=3 (small wavenumbers) and 7 (large wavenumbers). A simi- lar behavior has previously been reported in Ref.[15]
where it was speculated that the wavenumber value at the transition between the two power-laws should cor- respond to a characteristic spatial scale in the plasma.
The only length scale close to the transitional value was found to be the ion Larmor radiusρi.
The spectrum at small wavenumbers is roughly consistent with the inverse energy cascade in 2D turbulence, F2(k) ∝ k−8/3. The exponent at large wavenumbers does not fit into this framework. How- ever, for very large wavenumbers one enters the dissi- pation range; here, it has been argued that the energy spectrum could have one of the following dependen- cies
Fig. 1. Wavenumber spectrum of broadband turbulence in W7-AS measured using the LOTUS diagnostic. Squares are measured points. Solid lines are power-law fits to the three smallest and five largest wavenumbers, the dashed line is a fit to Eq.(1). The power-law fit grouping of points is the only one where convergence is obtained.
(5) ENeumann(k)∝e−ak, EHeisenberg(k)∝k−7, where a >0 is a constant having a dimension of length (see Ref.[12]and references therein). The en- ergy spectrum proposed by J. von Neumann was what initially inspired us to investigate an exponential de- cay of P (k) in Ref.[6]. Fitting all wavenumbers to Eq.(1),ENeumann(k)/A2, we find thatn=0.1 cm or a wavenumber of 55 cm−1. Alternatively, the transi- tional wavenumber found at the separation between the two power-laws is 31 cm−1(0.20 cm). The expres- sionEHeisenberg(k)/A2yieldsm=8, which is close to the experimental valuem=7 for large wavenumbers.
Calculating the ion Larmor radius at the electron tem- peratureρs for this case we find that it is 0.1 cm, i.e., the same order of magnitude as the spatial scales found above. We usedρsinstead ofρi because ion tempera- ture measurements were unavailable.
Currently we can think of three possible explana- tions for the behavior of the wavenumber spectrum:
(1) We observe 2D turbulence and the transition be- tween the two power-laws occurs at a spatial scale where the inverse energy cascade develops into the dissipation range. However, the enstrophy cas- cade is not accounted for in this case.
(2) We observe 2D turbulence in the dissipation range described by a single exponential function as pro- posed by J. von Neumann.
(3) Turbulence theory does not apply. The transi- tion between two power-laws or the characteris- tic scale found using a single exponential func- tion (Eq.(2)) indicates that one scale dominates the turbulent dynamics in the wavenumber range studied.
3. Galaxies 3.1. Inflation
Dramatic developments have taken place in cos- mology over the last decade, lending increasing sup- port to the paradigm of inflation as an explanation for what took place before the events described by the big bang theory [16]. Inflation solved the so-called hori- zon and flatness problems, but was at odds with earlier observations indicating that the ratio of the mass den- sity of the universe to the critical value, the density parameterΩ, was 0.2–0.3, while inflation predicted it should be 1:
Ω= ρ ρc,
(6) Ω <1: open, Ω=1: flat, Ω >1: closed, where ρc =3H02/8π G is the critical mass density, H0=70 km/s/Mpc is the Hubble parameter observed today andGis I. Newton’s gravitational constant[17].
However, new measurements in the late 1990’s lead to a drastic modification of Ω: Observations of type Ia supernovae (SN) showed that the separation velocity between galaxies was speeding up, not slowing down as would be expected for an open universe. The un- derlying explanation for this accelerated expansion is not understood, but it seems that the universe con- tains large quantities of negative pressure substance, creating a gravitational repulsion driving the expan- sion. This negative pressure material is called dark energy, the total density of dark energy ΩΛ is 0.7.
The existence of dark energy is equivalent to the cos- mological constantΛintroduced by A. Einstein. The dark matter densityΩd is 0.25 and the baryonic mat- ter density Ωb is 0.05, so the total density is very close (or equal) to the critical density. The SN Ia data is supported by measurements of nonuniformities in the cosmic microwave background (CMB) radiation.
The CMB anisotropy is due to the presence of tiny pri-
Fig. 2. Wavenumber spectrum of galaxies measured by the SDSS Team. Squares are measured points. Solid lines are power-law fits, the dashed line is a fit to Eq.(1). The power-law fit grouping of points is chosen so the combined, normalized,χ2of the fits is min- imized. The data is taken from Ref.[23].
mordial density fluctuations at the time of recombina- tion, where atoms formed. At that point in time the age of the universe was about 300 000 years and the tem- perature was 3000 K. The structures observed in the CMB are called acoustic peaks, and the simplest ver- sions of inflation all reproduce these structures quite accurately. The acoustic peaks cannot be modelled by assuming that the universe is open.
3.2. Wavenumber spectrum
A study of density fluctuations on large scales us- ing 205 443 galaxies has been published by the SDSS Team in Ref.[18], seeFig. 2. 3D maps of the universe are provided by the SDSS galaxy redshift survey, ob- serving about a quarter of the celestial sphere using a 2.5 m telescope and a charge-coupled device (CCD) camera. The galaxies had a mean redshift z≈0.1, corresponding to light emitted 1–2 Gyr ago[17]. Fix- ing some cosmological parameters to Wilkinson Mi- crowave Anisotropy Probe (WMAP) satellite values [19–21]one finds—using physics based models—that the wavenumber spectrum measurements were fitted by a matter densityΩm=Ωd+Ωb=0.295±0.0323.
In this caseh=H0/(100 km/s/Mpc)=0.72 was as- sumed and it was observed that the wavenumber spec- trum was not well characterized by a single power-law.
A follow-up paper by the SDSS Team, Ref.[22], combined non-CMB measurements (SDSS) with CMB measurements (WMAP) to constrain free pa-
rameters of cosmological models and break CMB de- generacies in parameter space. This resulted inΩm= 0.30±0.04 andh=0.70+−0.040.03. Adding the SDSS in- formation more than halved WMAP-only error bars on some parameters, e.g., the Hubble parameter and matter density.
The data presented inFig. 2 has been taken from M. Tegmark’s homepage[23]. According to the rec- ommendation by the SDSS Team [18], the three largest wavenumbers shown are not used in the fits described below.
As we did for the W7-AS data in Section2, we fit the SDSS measurements to two power-laws (Eq.(3)) or a single exponential function (Eq.(1)). The power- law fits are shown as solid lines, the exponential fit as a dashed line.
The power-law fits yield exponentsm= 0.8 (small wavenumbers) and 1.4 (large wavenumbers). The wavenumber ranges were determined by minimizing the combined, normalized,χ2of the fits. As the SDSS Team found, a single power-law cannot describe the observations. The exponents are not close to the ones governing fluid turbulence discussed in Section2. The transitional wavenumber is 0.09 h Mpc−1, correspond- ing to a length of 67 h−1Mpc.
We find the characteristic length from an expo- nential fit to be n=2 h−1Mpc or a wavenumber of 3 h Mpc−1.
4. Discussion
The fact that density fluctuations on small (fusion plasmas) and large (galaxies) scales can be described by an exponential function might indicate that plasma turbulence at early times has been expanded to cos- mological proportions. A natural consequence of that thought would be to investigate fluctuations in quark–
gluon plasmas (QGPs) corresponding to even earlier times. However, experimental techniques to do this are not sufficiently developed at the moment due to the ex- treme nature of QGPs.
It is fascinating that wavenumber spectra over wider scales peak at small wavenumbers and decrease both above and below that peak. This is seen both in fusion plasmas[24]and for galaxies, see, e.g., Fig. 38 in Ref.[18]. Turbulence theory in 1D or 3D would in- terpret the peak position as the scale where energy is injected.
Fitting wavenumber spectra to power-laws is based on fluid turbulence theories, but in general care must be taken when interpreting the outcome: We know that an exponential function can be Taylor expanded to an infinite power series:
(7) P (k)∝e−nk=∞
i=0
(−nk)i i! .
So locally, i.e., for a small range of wavenumbers, an exponential dependency can be masked as a power- law; the exponent would vary as a function of the wavenumber range selected.
We favor the exponential functions over power- laws as descriptors of the data, either including an algebraic prefactor as in Eq.(1) or using a pure ex- ponential function as in Eq.(2): In the latter case, fits yieldn=0.2 cm for fusion plasmas and 11 h−1Mpc for galaxies. The exponential decrease of density fluc- tuation power versus wavenumber implies that either (2) or (3) in Section2could explain the data. Perhaps forcing occurs at a large scale and transitions either di- rectly to dissipation (2) or to some effect governed by the scale determined from the exponential function (3) at smaller scales. The fact that we obtainnρs using Eq.(1)for fusion plasmas supports this conjecture.
5. Conclusions
We have in this Letter reported on suggestive sim- ilarities between density fluctuation power versus wavenumber on small (mm) and large (Mpc) scales.
The small scale measurements were made in fusion plasmas and compared to predictions from turbulence theory. The data fit Eq.(1), which is consistent with the dissipative range of 2D turbulence. Alternatively, the results fit Eq.(2)which has a functional form that cannot be explained by turbulence theory.
The large scale measurements were part of the SDSS galaxy redshift survey. As is the case for fusion plasmas, the galaxy data can be described by Eqs.(1) and (2). The similar dependency of density fluctuation power on wavenumber might indicate a common ori- gin of these fluctuations, perhaps from fluctuations in QGPs at early stages in the formation of the universe.
The cross-disciplinary work presented here is hope- fully just the beginning of an interesting path that can
benefit both fields. As a first step, we will expand our studies to encompass a wider range of scales, both for fusion plasma and galaxy measurements.
Acknowledgements
This work was supported at MIT by the Depart- ment of Energy, Cooperative Grant No. DE-FC02- 99ER54512.
References
[1] F.L. Hinton, R.D. Hazeltine, Rev. Mod. Phys. 48 (1976) 239.
[2] A.J. Wootton, et al., Phys. Fluids B 2 (1990) 2879.
[3] H. Renner, W7-AS Team, NBI Group, ICF Group, ECRH Group, Plasma Phys. Controlled Fusion 31 (1989) 1579.
[4] M. Saffman, et al., Rev. Sci. Instrum. 72 (2001) 2579.
[5] N.P. Basse, Ph.D. Thesis, University of Copenhagen, 2002, available onhttp://www.risoe.dk/rispubl/ofd/ris-r-1355.htm.
[6] N.P. Basse, et al., Phys. Plasmas 9 (2002) 3035.
[7] P. Hennequin, et al., Plasma Phys. Controlled Fusion 46 (2004) B121.
[8] M.C. Andersen, KVANT 14 (2003) 25.
[9]http://www.sdss.org/.
[10] U. Frisch, Turbulence, Cambridge Univ. Press, Cambridge, UK, 1995.
[11] G. Antar, et al., Plasma Phys. Controlled Fusion 40 (1998) 947.
[12] J. von Neumann, in: A.H. Taub (Ed.), Collected Works VI:
Theory of Games, Astrophysics, Hydrodynamics and Meteo- rology, Pergamon Press, Oxford, 1963.
[13] H. Tennekes, J.L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, 1972.
[14] G. Antar, Ph.D. Thesis, École Polytechnique, 1996.
[15] C. Honoré, et al., in: Proceedings of 25th European Conference on Controlled Fusion and Plasma Physics, Prague, 1998, in:
European Physical Society, vol. 22C, Petit-Lancy, Switzerland, 1998, p. 647.
[16] A.H. Guth, D.I. Kaiser, Science 307 (2005) 884.
[17] W.H. Kinney, astro-ph/0301448.
[18] M. Tegmark, et al., Astrophys. J. 606 (2004) 702.
[19] C.L. Bennett, et al., Astrophys. J. Suppl. Ser. 148 (2003) 1.
[20] D.N. Spergel, et al., Astrophys. J. Suppl. Ser. 148 (2003) 175.
[21] L. Verde, et al., Astrophys. J. Suppl. Ser. 148 (2003) 195.
[22] M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501.
[23] M. Tegmark’s homepage ishttp://space.mit.edu/home/tegmark and the SDSS data used is available onhttp://space.mit.edu/
home/tegmark/sdsspower/sdss_measurements.txt.
[24] N.P. Basse, et al., Characterization of core and edge turbulence in L- and enhanced Dα H-mode Alcator C-Mod plasmas, Phys. Plasmas (2005), in press.
